# Conical pendulum in rotating frame

• StrangelyQuarky
In summary, the question involves comparing the periods of a pendulum moving to the east and west at an angle to the vertical at the north pole. The equation of motion in the rotating frame includes the Coriolis force and a modified gravitational acceleration due to the Earth's rotation. The pendulum moving east has a longer period due to the need to "catch up" with the rotating ground. To calculate the periods, the problem can be analyzed from a nonrotating frame or by using a free body diagram in the rotating frame.
StrangelyQuarky

## Homework Statement

A pendulum of length $l$ at the north pole is moving in a circle to the east at an angle $\theta$ to the vertical. It has some period $T_E$ as measured in the rotating Earth frame. The experiment is then repeated except now the pendulum is moving to the west with period $T_W$. The question asks which period is longer, and to calculate the relative time difference in the periods.

## Homework Equations

In the rotating frame, the equation of motion of the pendulum involves the Coriolis force and a modified gravitational acceleration due to the rotation of the Earth: $$m\mathbf{a'} = m\mathbf{g'} + \mathbf{T} -2m(\mathbf{\omega '} \times \mathbf{v'})$$

where T is the tension in the wire.

## The Attempt at a Solution

Since the Earth rotates counterclockwise as viewed looking down on the north pole, the pendulum that moves to the east has a longer period because the Earth's rotation means it has to "catch up" with the rotating ground (I think?). However, I am at a loss as to how to calculate the period from equation of motion.

StrangelyQuarky said:
In the rotating frame, the equation of motion of the pendulum involves the Coriolis force and a modified gravitational acceleration due to the rotation of the Earth: $$m\mathbf{a'} = m\mathbf{g'} + \mathbf{T} -2m(\mathbf{\omega '} \times \mathbf{v'})$$
Can you describe in more detail the "modified gravitational acceleration" ##\mathbf{g'}##? What is its magnitude and direction?

What does the symbol ##\mathbf{\omega '}## stand for?

What about the "centrifugal force" in the rotating frame? Or does your ##\mathbf{g'}## include that?

Since the Earth rotates counterclockwise as viewed looking down on the north pole, the pendulum that moves to the east has a longer period because the Earth's rotation means it has to "catch up" with the rotating ground (I think?).
Yes, the "catching up" is from the point of view of an observer in a nonrotating. In fact, you can work the problem by analyzing it from the nonrotating frame. But, if you want to work it out in the rotating frame, then follow the usual recipe of starting with a free body diagram, etc.

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## 1. What is a conical pendulum in a rotating frame?

A conical pendulum in a rotating frame is a pendulum that is attached to a pivot point that is itself rotating in a circular motion. This creates a cone-shaped path for the pendulum bob to follow, rather than a straight back-and-forth motion like a traditional pendulum in a stationary frame.

## 2. How does the rotation of the frame affect the motion of the pendulum?

The rotation of the frame causes the pendulum bob to experience a centrifugal force, which pulls it outwards from the center of the circle. This force, along with the tension force from the string, creates a conical motion for the pendulum.

## 3. What factors affect the period of a conical pendulum in a rotating frame?

The period of a conical pendulum in a rotating frame is affected by the length of the string, the mass of the pendulum bob, the angular velocity of the frame, and the angle of the pendulum with respect to the frame's axis of rotation.

## 4. Can a conical pendulum in a rotating frame exhibit simple harmonic motion?

No, a conical pendulum in a rotating frame does not exhibit simple harmonic motion because the restoring force (tension in the string) is not directly proportional to the displacement from the equilibrium position. However, for small angles of displacement, it can exhibit approximately simple harmonic motion.

## 5. What are some real-life applications of conical pendulums in rotating frames?

Conical pendulums in rotating frames can be found in various devices such as Foucault pendulums, which demonstrate the rotation of the Earth, and gyroscopes, which are used in navigation systems. They are also used in amusement park rides and in certain types of clocks.

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